(a) Evaluate the integral as the limit of a sum ∫₁² (2x+1)dx OR (b) Find the probability of guessing correctly

Question 27 of 42

\int_1^2 (2x+1)\,dx

(OR)

(b) Find the probability of guessing correctly atleast six of the ten answers in a TRUE/FALSE objective test.

Puducherry TnboardTamil Nadu HSC (DGE) Commerce Board 2023Subjective· 5mImportance★★★★★

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Limit-of-a-sum gives $\int_1^2(2x+1)dx=4$ . (b) $P(X\ge6)=\dfrac{386}{1024}\approx0.377$ . (a) Evaluate $\displaystyle\int_1^2(2x+1)\,dx$ as the limit of a sum. Use $\displaystyle\int_a^b f(x)\,dx=\lim_{n\to\infty}h\sum_{r=1}^{n}f(a+rh)$ with $a=1,\ b=2,\ h=\dfrac{b-a}{n}=\dfrac1n$ . $f(a+rh)=2(1+rh)+1=3+2rh.$ $h\sum_{r=1}^{n}(3+2rh)=h\Big[3n+2h\cdot\frac{n(n+1)}{2}\Big]=3nh+h^{2}n(n+1).$ Put $h=\dfrac1n$ : $\;3n\cdot\dfrac1n+\dfrac{1}{n^{2}}\,n(n+1)=3+\dfrac{n+1}{n}=3+1+\dfrac1n.$ $\lim_{n\to\infty}\Big(4+\frac1n\Big)=4.$ (Verification: $[x^{2}+x]_1^2=6-2=4$ .)
Guessing correctly at least 6 of 10 TRUE/FALSE answers. …

Q.(a) Evaluate the integral as the limit of a sum $\int_1^2 (2x+1)\,dx$